Simple Polyadic Groups H. Khodabandeh, M. Shahryari AAD May 2012 - - PowerPoint PPT Presentation

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups

• H. Khodabandeh, M. Shahryari

AAD

May 2012

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

A simple notation

During this presentation, we use the following notations:

• 1. Any sequence of the form xi, xi+1, . . . , xj will be denoted by

xj

i

• 2. The notation

(t)

x will denote the sequence x, x, . . . , x (t times). So if G is a set and f : Gn → G is a function, we can denote the element f(x1, x2, . . . , xn) by f(xn

1).

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

A polyadic group is . . .

a non-empty set G together with an n-ary operation f : Gn → G such that

• 1. The operation f is associative, i.e.

f(xi−1

1

, f(xn+i−1

i

), x2n−1

n+i ) = f(xj−1 1

, f(xn+j−1

j

), x2n−1

n+j ),

where 1 ≤ i, j ≤ n, and x1, . . . , x2n−1 ∈ G.

• 2. For fixed a1, a2, . . . , an, b ∈ G and all i ∈ {1, . . . , n}, the

following equations have unique solutions for x; f(ai−1

1

, x, an

i+1) = b.

We denote the polyadic group by (G, f). More precisely, we call (G, f) an n-ary group.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Examples of polyadic groups

Suppose (G, ◦) is an ordinary group and define f(xn

1) = x1 ◦ x2 ◦ · · · ◦ xn.

Then (G, f) is polyadic group which is called of reduced type. We write (G, f) = dern(G, ◦).

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Example ...

Suppose (G, ◦) is an ordinary group and b ∈ Z(G). Define f(xn

1) = x1 ◦ x2 · · · ◦ xn ◦ b.

Then (G, f) is polyadic group which is called b-derived polyadic group from G and it is denoted by dern

b (G, ◦).

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Example ...

Suppose G = Sm \ Am, (the set of all odd permutations of degree m). Then by the ternary operation f(x1, x2, x3) = x1x2x3 the set G is a ternary group.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Example ...

Suppose ω is a primitive n − 1-th root of unity in a field K. Let G = {x ∈ GLm(K) : det x = ω}. Then G is an n-ary group by the operation f(xn

1) = x1x2 · · · xn.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Identity in polyadic groups

An n-ary group (G, f) is of reduced type iff it contains an element e (called an n-ary identity) such that f(

(i−1)

e , x,

(n−i)

e ) = x holds for all x ∈ G and i = 1, . . . , n.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Skew element

From the definition of an n-ary group (G, f), we can directly see that for every x ∈ G, there exists only one z ∈ G satisfying the equation f(

(n−1)

x , z) = x. This element is called skew to x and is denoted by x.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Retracts of polyadic groups

Let (G, f) be an n-ary group and a ∈ G be a fixed element. Define a binary operation on G by x ∗ y = f(x,

(n−2)

a , y). It is proved that (G, ∗) is an ordinary group, which we call the retract of G over a. The notation for retract: Reta(G, f), or simply by Reta(G). Retracts of a polyadic group are isomorphic.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

The identity and inverse

The identity of the group Reta(G) is a. The inverse element to x has the form x−1 = f(a,

(n−3)

x , x, a).

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Recovering a polyadic group from its retracts

Any n-ary group can be uniquely described by its retract and some automorphism of this retract. Theorem Let (G, f) be an n-ary group. Then

• 1. on G one can define an operation · such that (G, ·) is a group,
• 2. there exist an automorphism θ of (G, ·) and b ∈ G, such that

θ(b) = b,

• 3. θn−1(x) = bxb−1, for every x ∈ G,
• 4. f(xn

1) = x1θ(x2)θ2(x3) · · · θn−1(xn)b, for all x1, . . . , xn ∈ G.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Remark

According to this theorem, we use the notation derθ,b(G, ·) for (G, f) and we say that (G, f) is (θ, b)-derived from the group (G, ·). The binary group (G, ·) is in fact Reta(G, f). We will assume that (G, f) = derθ,b(G, ·).

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Normal subgroups

An n-ary subgroup H of a polyadic group (G, f) is called normal if f(x,

(n−3)

x , h, x) ∈ H for all h ∈ H and x ∈ G.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

GTS

If every normal subgroup of (G, f) is singleton or equal to G, then we say that (G, f) is group theoretically simple or it is GTS for short. If H = G is the only normal subgroup of (G, f), then we say it is strongly simple in the group theoretic sense or GTS∗ for short.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

UAS

An equivalence relation R over G is said to be a congruence, if

• 1. ∀i : xiRyi ⇒ f(xn

1)Rf(yn 1 ),

• 2. xRy ⇒ xRy.

We say that (G, f) is universal algebraically simple or UAS for short, if the only congruence is the equality and G × G.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Quotients are reduced

Theorem Suppose H (G, f) and define R =∼H by x ∼H y ⇔ ∃h1, . . . , hn−1 ∈ H : y = f(x, hn−1

1

). Then R is a congruence and if we let xH = [x]R, (the equivalence class of x), then the set G/H = {xH : x ∈ G} is an n-ary group with the operation fH(x1H, . . . , xnH) = f(xn

1)H.

Further we have (G/H, fH) = der(retH(G/H, fH)), and so it is reduced.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

UAS is also GTS

Theorem Every UAS is also GTS. But the converse is not true!

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Facts about congruences

Cong(G, f) is the set of all congruences of (G, f). This set is a lattice under the operations of intersection and product (composition). We also denote by Eq(G) the set of all equivalence relations of G. Theorem R ∈ Cong(G, f) iff R ∈ Eq(G) and R is a θ-invariant subgroup

• f G × G.

Corollary We have Cong(G, f) = {R ≤θ G × G : ∆ ⊆ R}.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

UAS

Theorem (G, f) is UAS iff the only normal θ-invariant subgroups of (G, ·) are trivial subgroups.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Structure of normals

For u ∈ G, define a new binary operation on G by x ∗ y = xu−1y. Then (G, ∗) is an isomorphic copy of (G, ·) Theorem We have H (G, f) iff there exists an element u ∈ H such that

• 1. H is a ψu-invariant normal subgroup of Gu,
• 2. for all x ∈ G, we have θ−1(x−1u)x ∈ H.
• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

GTS

Theorem A polyadic group (G, f) is GTS∗ iff whenever K is a θ-invariant normal subgroup of (G, ·) with θK inner, then K = G.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Example

Example Let (G, ·) be a non-abelian simple group and θ be an automorphism of order n − 1. Then derθ(G, ·) is a UAS n-ary group. The number of non-isomorphic polyadic groups of the form derθ(G, ·) is the same as the number of conjugacy classes of Out(G), the group of outer automorphisms of (G, ·)

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Example

Example Suppose p is a prime and G = Zp × Zp. Let q(t) = t2 + at + b be an irreducible polynomial over the field Zp and choose a matrix A ∈ GL2(p) with the characteristic polynomial q(t). Let An−1 = I and define an automorphism θ : G → G by θ(X) = AX. Clearly, θ has no non-trivial invariant subgroup, since q(t) is irreducible. So, derθ(G, ·) is a UAS n-ary group. Note that, we have f(Xn

1 ) = X1 + AX2 + · · · + An−2Xn−1 + Xn.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Example

Example Let H be a non-abelian simple group with an outer automorphism θ. Let θn−1 = id and G = H × H. Then θ extends to G by θ(x, y) = (θ(x), θ(y)). The subgroups K1 = H × 1 and K2 = 1 × H are the only θ-invariant normal subgroups of G. Clearly θKi : G/Ki → G/Ki is not inner as we supposed θ an outer automorphism. Therefore derθ(G, ·) is a GTS polyadic group but it is not UAS.

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME

Simple Polyadic Groups C ¸ ES ¸ ME

Thank You! Thanks to The Participants . . . . . . . . . . . . . . . . . . .For Listening...

and

The Organizers . . . . For Taking Care of Everything...

• H. Khodabandeh, M. Shahryari

Simple Polyadic Groups C ¸ ES ¸ ME